CHAPTER 41
Functions of Several Variables
MULTIVARIATE FUNCTIONS AND THEIR GRAPHS
41.1 Sketch the cylinder
See Fig. 41-1. The surface is generated by taking the ellipse in the ry-plane and moving it
parallel to the z-axis (z is the missing variable).
Fig. 41-1 Fig. 41-2
41.2 Describe and sketch the cylinder z = y .
See Fig. 41-2. Take the parabola in the _yz-plane z = y and move it parallel to the jt-axis (x is the missing
variable).
41.3 Describe and sketch the graph of 2x + 3>y = 6.
See Fig. 41-3. The graph is a plane, obtained by taking the line 2x + 3y = 6, lying in the *.y-plane, and
moving it parallel to the z-axis.
Fig. 41-3 Fig. 41-4
41.4 Write the equation for the surface obtained by revolving the curve z = y2 (in the yz-plane) about the z-axis.
When the point (0, y*, z*) on z - y2 is rotated about the z-axis (see Fig. 41-4), consider any resulting
point (x, y, z). Clearly, z = z* and Hence, * +y" = (y*) = z* = z. So, the result-
ing points satisfy the equation z = x + y2.
361
, 362 CHAPTER 41
41.5 Write an equation for the surface obtained by rotating a curve f ( y , z) = 0 (in the yz-plane) about the z-axis.
This is a generalization of Problem 41.4. A point (0, y*, z*) on the curve yields points (x, y, z), where
z = z* and Hence, the point (x, y, z) satisfies the equation
41.6 Write an equation for the surface obtained by rotating the curve (in the yz-plane) about the
z-axis.
By Problem 41.5, the equation is obtained by replacing y by in the original equation. So, we get
an ellipsoid.
41.7 Write an equation of the surface obtained by rotating the hyperbola (in the ry-plane) about the
*-axis.
By analogy with Problem 41.5, we replace y by obtaining
1. (This surface is called a hyperboloid of two sheets.)
41.8 Write an equation of the surface obtained by rotating the line z = 2y (in the yz-plane) about the z-axis.
By Problem 41.5, an equation is This is a cone (with both nappes) having
the z-axis as axis of symmetry (see Fig. 41-5)
Fig. 41-5 Fig. 41-6
41.9 Write an equation of the surface obtained by rotating the parabola z = 4 — x2 (in the xz-plane) about the
z-axis. (See Fig. 41-6).
By Problem 41.5, an equation is z = 4 —i z = 4 - (x2 + y2). This is a circular paraboloid.
41.10 Describe and sketch the surface obtained by rotating the curve z = \y\ (in the yz-plane) about the z-axis.
By Problem 41.5. an equation is which is equivalent to z 2 = x2 + y2 for z a 0. This is
a right circular cone (with a 90° apex angle); see Fig. 41-7.
Fig. 41-7
Functions of Several Variables
MULTIVARIATE FUNCTIONS AND THEIR GRAPHS
41.1 Sketch the cylinder
See Fig. 41-1. The surface is generated by taking the ellipse in the ry-plane and moving it
parallel to the z-axis (z is the missing variable).
Fig. 41-1 Fig. 41-2
41.2 Describe and sketch the cylinder z = y .
See Fig. 41-2. Take the parabola in the _yz-plane z = y and move it parallel to the jt-axis (x is the missing
variable).
41.3 Describe and sketch the graph of 2x + 3>y = 6.
See Fig. 41-3. The graph is a plane, obtained by taking the line 2x + 3y = 6, lying in the *.y-plane, and
moving it parallel to the z-axis.
Fig. 41-3 Fig. 41-4
41.4 Write the equation for the surface obtained by revolving the curve z = y2 (in the yz-plane) about the z-axis.
When the point (0, y*, z*) on z - y2 is rotated about the z-axis (see Fig. 41-4), consider any resulting
point (x, y, z). Clearly, z = z* and Hence, * +y" = (y*) = z* = z. So, the result-
ing points satisfy the equation z = x + y2.
361
, 362 CHAPTER 41
41.5 Write an equation for the surface obtained by rotating a curve f ( y , z) = 0 (in the yz-plane) about the z-axis.
This is a generalization of Problem 41.4. A point (0, y*, z*) on the curve yields points (x, y, z), where
z = z* and Hence, the point (x, y, z) satisfies the equation
41.6 Write an equation for the surface obtained by rotating the curve (in the yz-plane) about the
z-axis.
By Problem 41.5, the equation is obtained by replacing y by in the original equation. So, we get
an ellipsoid.
41.7 Write an equation of the surface obtained by rotating the hyperbola (in the ry-plane) about the
*-axis.
By analogy with Problem 41.5, we replace y by obtaining
1. (This surface is called a hyperboloid of two sheets.)
41.8 Write an equation of the surface obtained by rotating the line z = 2y (in the yz-plane) about the z-axis.
By Problem 41.5, an equation is This is a cone (with both nappes) having
the z-axis as axis of symmetry (see Fig. 41-5)
Fig. 41-5 Fig. 41-6
41.9 Write an equation of the surface obtained by rotating the parabola z = 4 — x2 (in the xz-plane) about the
z-axis. (See Fig. 41-6).
By Problem 41.5, an equation is z = 4 —i z = 4 - (x2 + y2). This is a circular paraboloid.
41.10 Describe and sketch the surface obtained by rotating the curve z = \y\ (in the yz-plane) about the z-axis.
By Problem 41.5. an equation is which is equivalent to z 2 = x2 + y2 for z a 0. This is
a right circular cone (with a 90° apex angle); see Fig. 41-7.
Fig. 41-7
